A refined model of elastic-plastic bending deformation of flexible reinforced shallow shells based on explicit “cross” scheme

An initial-boundary value problem is formulated for elastoplastic deformation of flexible shallow shells, cross reinforced by equidistant surfaces. The mechanical behavior of the component materials of the composition of curved panels is governed by the Prandtl-Reuss-Hill equations for an elastoplastic medium. The geometric nonlinearity of the problem is considered in the Karman approximation. The resulting governing equations and the corresponding initial and boundary conditions in the generalized kinematic variables allow one to calculate with different accuracy the stress-strain state in the components of the composition of flexible shallow shells and plates, taking into account their weakened resistance to the transverse shear. The relations corresponding to the traditional non-classical Reddy theory are obtained in the first approximation from the equations, initial and boundary conditions. The numerical integration of the initial-boundary value problem is carried out on the basis of the method of steps in time. The central finite differences are used to approximate derivatives with respect to time. On the basis of this approximation the explicit numerical “cross” scheme is constructed in the case of impact loads of explosive type. The properties of elastoplastic dynamic behavior is investigated for the reinforced shallow spherical shell of annular form in plan, with a perfectly rigid inner insert, also for the cylindrical panels of rectangular elongated shape in plan of different thickness under the action of the front load generated by the air blast. Shallow shells are rationally reinforced in the directions of principal stresses and strains and at the supported edges they are rigidly clamped. It is shown that in some cases the Reddy theory is absolutely inadmissible to produce adequate results of calculations of the elastic-plastic deformable composite shallow shells, even with a relatively small thickness. It is demonstrated that, due to the geometrical and physical nonlinearity of the investigated problem, the dynamic behavior of reinforced curved panels significantly depends on the fact that the form of the front surface of the shell, subjected to the external load, is convex or concave. It is established that the most probable mechanism of pre-destruction of such structures is the accumulation of damage due to low-cycle fatigue of binder that occurs in the oscillation process of a reinforced structure.